What is the integral of x(sqrt(x)-1/(x^2)) ?

The integral of x(sqrt(x)-1/(x^2)) is -ln|x|+2/5 x^{5/2}+C

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integral of x(sqrt(x)-1/(x^2))

\int x (x - \frac{1}{x ^{2}}) d x

- ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

\int x (x - \frac{1}{x ^{2}}) d x

Expand

x (x - \frac{1}{x ^{2}}) : - \frac{1}{x} + x^{\frac{3}{2}}

= \int - \frac{1}{x} + x^{\frac{3}{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{x} d x + \int x^{\frac{3}{2}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

= - ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= - ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}} + C

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What is the integral of x(sqrt(x)-1/(x^2)) ?
The integral of x(sqrt(x)-1/(x^2)) is -ln|x|+2/5 x^{5/2}+C